If a point has no dimension and no area how can there be space? Everyone knows that a point on any plane has zero magnitude, height, width, or volume. A line is a collection of points, and a plane is a collection of lines, while a cube is a collection of planes and so forth.
How is it then that any line, surface or cube can be made of points that have no magnitude area etc.
 A: In order to calculate the "sum" of lengths/areas/volumes etc. for a collection of subsets, you need the collection of subsets to be countable. But if you take a curve or surface etc. and look at all the points on the set, it is uncountable. So you can't reason that the total length/area/volume etc. is $0 + 0 + \ldots = 0$. If anything, you have to consider that e.g. for length of $S$ along the line you get $dx + dx+ \ldots$, "summed" over all points in $S$, which is indeed zero for a countable point set $S$ but for all points on a line segment you get an integral (not countable sum) of $dx$, which gives the length of the line segment.
A: Because length, area, volume, etc. are not ways of counting points, but ways of counting sizes relative to a unit. The structures of a topological space or measure space assign the property of being a blob of space to collections of points, rather than assigning a makeup of points to them. When you have a finite set of points in what you want to be blobs of space, the counting measure is appropriate. But not when you get to infinite sets. Consider the map that sends the $n^{\text{th}}$ odd number to $n$ and the $n^{\text{th}}$ even number to $-n$, starting from the $0^{\text{th}}$ even number. Clearly the number of points is equal between the natural numbers and the integers. But you might find it more appropriate for the natural numbers to be half the size of the integers. Furthermore, any two intervals, by number of points, would be equal in length if you were measuring space as the number of points you can count in a region. The point is, if you're axiomatizing your notion of relative size, that's not the same as acquiring a notion of point content where every point contributes the same amount to the size.
A: The issue you raise here is similar/related to things like:
Archimedes axiom:
http://en.wikipedia.org/wiki/Archimedean_axiom
and Zeno's Paradoxes
http://en.wikipedia.org/wiki/Zeno%27s_paradoxes
A: I think the idea is that lines are sets of points. The length of a line is equal to the cardinality of the set of the points that comprise it. 
Said otherwise, let line segment $P_{1}P_{3}$ designate the set $\langle P_1,P_2,P_3 \rangle$. The line segment is 3 units long. 
In this way, assigning actual measures, such as inches, to units isn't any different than assigning actual things to numbers. For example, $2$ apples. 
Math is abstract. Abstraction involves leaving things out (alternatively, concretion involves adding things in; however you prefer). Leaving out physical things allows us to generalize, which is a reason that math is so broadly applicable. 
